CHAPTER IV. 



MOTION OF A LIQUID IN TWO DIMENSIONS. 



68. IF the velocities u, v be functions of x, y only, whilst w 

 is zero, the motion takes place in a series of planes parallel to xy y 

 and is the same in each of those planes. The investigation of the 

 motion of a liquid under these circumstances is characterized by 

 certain analytical peculiarities ; and the solutions of several pro 

 blems of great interest are readily obtained. 



Since the whole motion is known when we know that in the 

 plane z = 0, we shall confine our attention to the motion which 

 takes place in that plane. When we speak of points and lines 

 drawn in that plane, we shall in general understand them to re 

 present respectively the straight lines parallel to the axis of z, 

 and the cylindrical surfaces having their generating lines parallel 

 to the axis of z t of which they are the traces. 



By the flux across any curve we shall understand the volume 

 of fluid which in unit time crosses that portion of the cylindrical 

 surface having the curve as base, which is included between the 

 planes z = 0, z \. 



69. Let A, P be any two points in the plane ocy. The flux 

 across any two lines joining AP is the same, provided they can be 

 reconciled without passing out of the region occupied by the 

 moving liquid ; for otherwise the space included between these 

 two lines would be gaining or losing fluid. Hence if A be 

 fixed, and P variable, the flux across any line AP is a 



